Global bifurcations and chaotic dynamics for a string-beam coupled system

The global bifurcations and chaotic dynamics of a string-beam coupled system subjected to parametric and external excitations are investigated in detail in this paper. The governing equations are firstly obtained to describe the nonlinear transverse vibrations of the string-beam coupled system. The Galerkin procedure is introduced to simplify the governing equations of motion to ordinary differential equations with two-degrees-of-freedom. Using the method of multiple scales, parametrically and externally excited system is transformed to the averaged equation. The case of 1:2 internal resonance between the modes of the beam and string, principal parametric resonance for the beam and primary resonance for the string is considered. Based on the averaged equation, the theory of normal form is utilized to find the explicit formulas of normal form associated with one double zero and a pair of pure imaginary eigenvalues. The global perturbation method is employed to analyze the global bifurcations and chaotic dynamics of the string-beam coupled system. The analysis of the global bifurcations indicates that there exist the homoclinic bifurcations and the Silnikov type single-pulse homoclinic orbit in the averaged equation of the string-beam coupled system. These results obtained here mean that the chaotic motions can occur in the string-beam coupled system. Numerical simulations also verify the analytical predications.